Modal- μ Definable Graph Transduction

1. Modal-μ DefinableGraph Transduction Kazuhiro InabaNational Institute of Informatics, Japan 4th DIKU-IST Workshop, 2011

2. What We Want to Do • Verification of Graph-to-Graph Transformations • e.g., Queries on Graph-Structured Database or Transformations of XML with “id” links a A f b B a A B b

3. What We Want to Do • Verification of Graph-to-Graph Transformationswith respect to input/output specifications φIN “From (a) we can reach (a) again.” φOUT “From (A) we can reach (A) again.” a A f b B a A B b verify whether or not: for any graph G, G ⊧ φIN⇒ f(G)⊧ φOUT

4. Verification by Pre-Image(a.k.a. “weakest precondition” or “inverse type inference”) Given f and φOUT, compute invf(φOUT) such that: for any graph G, f(G)⊧ φOUT iff G ⊧ invf(φOUT) invf(φOUT) φOUT f φIN Then “for any graph G, G ⊧ φIN⇒ f(G)⊧ φOUT” iff “for any graph G, G ⊧ (φIN→ invf(φOUT ))” i.e., φIN→ invf(φOUT ) is valid

5. To Be More Concrete… • Which logic can we use for specifying φIN/OUT ? • Must be strong enough to express useful conditions. • Must be weak enough to have decidable validity. • What kind of transformation f can be verified ? • We must be able to compute the pre-image.

6. Our Approach • Take Modal-μ Calculus as the specification logic • (At least for trees) capture all existing XML-Schemas • Define a new model of graph transformationcalled Modal-μ Definable Transduction • Pre-image of modal-μ sentence can befully automatically computed • Expressive enough to capture(unnested) structural recursion on graphs

7. Related Work • MSO (Monadic 2nd-Order Logic) Definable Transduction • Overall structure is more or less the same. • Ours is a proposal to use Modal-μ instead of MSO • Hoare-Style Verification of Imperative Programs • Ours don’t deal with pointers or destructive updates. • Rather, it is more suitable forfunctional programs • Structural recursion is handledwithout any annotations { φIN} p := root while p != null do q := p.next p.next := p.next.next p := q end { φOUT} fun f( {$l: $x} ) = {cap($l) : g($x)} fun g( {_: $x} ) = f($x) { φIN} f { φOUT}

8. Outline • Two Kinds of Logics on Graphs • Predicate Logics • Modal Logics • Why Modal-μ ? • Review: Predicate-Logic Based Approach • MSO-Definable Graph Transduction [Courcelle 94] • Our Work: • Modal-μ Definable Graph Transduction • Computation of Pre-Image

9. Graphs (in Today’s Talk) • Σ : Finite Nonempty Alphabet • G = (V, E, π) • V Set of Nodes • E ⊆ V × V Set of Directed Edges • π : V → 2Σ Labels on Nodes π = → {a,b} → {a} → {b} → {} Σ = {a, b} V = { } a b a b a a b b

10. Predicate Logics on Graphs φ ::= | False | ￢ φ | φ ∨ φ | σ(x) (for σ∈Σ) “node x is labeled σ” | edge(x, y) “an edge connects x to y” | ∃x. φ “there’s x that makes ψ hold” | ∃S. φ “there’s a set S that makes ψ hold” | x ∈ S “x is in S” MSO FO • We can define True, φ∧φ , φ→φ, ∀x.φ, and ∀S.φ.

11. Semantics • For a graph G=(V,E,π) and an environmentΓ : Var→V • G, Γ ⊧ σ(x) iff σ ∈ π(Γ(x)) “node x is labeled σ” • G, Γ ⊧ edge(x, y) iff (Γ(x), Γ(y)) ∈ E“an edge connects x to y” • G, Γ ⊧ ∃x.φ iff there’s v∈V s.t. G,Γ[x:v] ⊧ φ …

12. Modal Logics on Graphs ψ ::= | False | ￢ φ | φ ∨ φ | σ (for σ∈Σ) “current node is labeled σ” | ◇φ “current node has an outgoing edge whose destination satisfies φ” | X | μX.φ “least fixpoint”(X must be in even # of ￢) Mμ M • We Can Define: □φ (dual of ◇) and νX.φ (GreatestFixPt)

13. Semantics • For a graph G=(V,E,π), an environmentΓ : Var→2V, and the current node v ∈ V • G, v, Γ ⊧ σ iff σ ∈ π(v) “current node is labeled σ” • G, v, Γ ⊧ ◇φiff there’s w (v,w)∈E & G,w,Γ ⊧ φ “current node has an outgoing edge whose destination satisfies φ” • G, v, Γ ⊧ μY. φ iff v ∈ LFP(F) where F(A) = {w∈V | G, w, Γ[Y:A] ⊧ φ} …

14. Examples • “From the node x, we can reach a σ-node” ∀S. ( (x∈S ∧∀y.∀z.(y∈S ∧ (edge(y,z)→z∈S)))→∃y. (y ∈ S ∧ σ(y))) • “Confluence” ∀y. ∀z. ( edge(x,y) ∧ edge(x,z)→∃w. (edge(y,w) ∧ edge(z,w)) ) • “From the current node, we can reach a σ-node” μY. (σ ∨ ◇Y) • “Confluence” (No way to express it in Modal-μ)

15. MSO Definable (1-copying)Transduction [Courcelle 94] A set of MSO formulas T = ・ σOUT(x) for each σ∈Σ ・ edgeOUT(x,y) defines a transformation fT converting G = (V, E, π) into G’ = (V, E’, π’) where • π’( v ) = { σ | G, x:v ⊧ σOUT(x) } • E’ = { (v, w) | G, x:v, y:w ⊧ edgeOUT (x,y) }

16. Example (Σ = {a, b, A, B}) edgeOUT(x, y) ≡ ∃z.(edge(x,z)∧ edge(z,y)) aOUT(x) ≡ bOUT(x) ≡ False AOUT(x) ≡ a(x) BOUT(x) ≡ b(x) a A b B a A B b

17. Pre-Image is Easily Obtained edgeOUT(x, y) ≡ ∃z.(edge(x,z)∧ edge(z,y)) aOUT(x) ≡ bOUT(x) ≡ False AOUT(x) ≡ a(x) BOUT(x) ≡ b(x) a A b B a A B b ∀x. A(x) → ∃y. edge(x,y)∧ A(y) ∀x. a(x)→ ∃y.∃z.(edge(x,z) ∧ edge(z,y))∧ a(y) Inline Expansion

18. Expressiveness & Complexity MSO FO Modal Modal-μ ∃x.φ ◇φ PSPACE Undecidable EXPTIME ∃S.φ μX.φ

19. Expressiveness & Complexity(on “tree-like” graphs) MSO FO Modal Modal-μ ∃x.φ ◇φ PSPACE NonElementary EXPTIME ∃S.φ μX.φ

20. Modal-μ and MSO • Complexity of Validity Checking • Modal-μ : EXPTIME-complete • MSO : Undecidable (Even in Trees, HyperEXP) • Expressive Power • Modal-μ= Bisimulation-Invariant Subset of MSO[Janin & Walukiewicz 96] • “Bisimulation-Invariant” ≃ “Physical equality of pointers cannot be checked” • Not a severe restriction for purely functional programs!

21. Modal-μ Definable (1-copying)Transduction A set of Modal-μ formulas T = ・σOUT for each σ∈Σ ・ edgeOUT an existential formula Fv={X} defines a transformation fT converting G = (V, E, π) into G’ = (V, E’, π’) where • π’( v ) = { σ | G, v ⊧ σOUT } • E’ = { (v, w) | G, v, X:{w} ⊧ edgeOUT }

22. Example (Σ = {a, b, A, B}) edgeOUT≡ ◇◇XaOUT≡ bOUT≡ False AOUT≡ a BOUT≡ b a A b B a A B b A → ◇A a→ ◇◇a

23. Existential Formula • A formula e with one free variable X isexistential, if • Examples: • “X ∨ True” is not existential (Consider P={}). • “◇X” is existential. • “□X” is not (when v is a leaf node …). • “σ” is not, but “X ∧ σ” is. for all G=(V,E,π), v∈V, P⊆V G, v, X:P ⊧ e iff ∃w∈P. G, v, X:{w} ⊧ e

24. Syntactic Condition for all G=(V,E,π), v∈V, P⊆V G, v, X:P ⊧ e iff ∃w∈P. G, v, X:{w} ⊧ e • Theorem:e is existential if it is in the following syntax e ::= False | X | Y | e ∨ e | ◇e | μY. e | e ∧ φ where φ is any formula without free variables(True, ￢, σ, □, and GFP must be “guarded” by _ ∧ _) Open Question: is this a necessary condition ?(i.e., do all existential formulas have logically-equivalent forms in this syntax?)

25. More Examples (Non-Examples) • edgeOUT≡ X edgeOUT≡ a • edgeOUT≡ μY. ((X ∧ a) ∨ ◇Y) edgeOUT≡ X∧◇X • edgeOUT≡ μY. ((X ∧ a ∧ □b) ∨ (￢a ∧ ◇Y) a a a b b a a a b

26. Pre-Image Computation For T = (σOUT, eOUT), define • inv( False ) = False • inv( ￢ φ ) = ￢ inv( φ ) • inv( φ1∨ φ2 ) = inv( φ1 ) ∨ inv( φ2 ) • inv( σ ) = σOUT • inv( ◇ φ ) = edgeOUT [X / inv(φ)] • inv( Y ) = Y • inv( μY. φ ) = μY. inv(φ) Theorem: fT(G), v ⊧ φ iff G, v ⊧ inv(φ)

27. Proof of the Theorem • By Induction on φ. The essential case is: G, v ⊧ inv(◇φ) iff G,v⊧edgeOUT [X / inv(φ)] (definition of inv) iff ∃w (G,v,X:{w} ⊧edgeOUT and G,w ⊧inv(φ)) (ext) iff ∃w ((v,w) in E’ and G,w ⊧inv(φ)) (def of E’) iff ∃w ((v,w) in E’ and fT(G),w ⊧φ) (IH) iff fT(G), v ⊧ ◇φ(definition of ◇) Theorem: fT(G), v ⊧ φ iff G, v ⊧ inv(φ)

28. n-copyingModal-μ DefinableTransduction A set of Modal-μ formulas T = ・σkOUT for each σ∈Σ, k∈{1 .. n} ・ edgeikOUT for each i, k∈{1 .. n} : existential defines a transformation fT converting G = (V, E, π) into G’ = (V*{1..n}, E’, π’) where • π’( <v,k> ) = { σ | G, v ⊧ σkOUT } • E’ = { (<v,i>, <w,k>) | G, v, X:{w} ⊧ edgeikOUT }

29. Example (Σ = {a, b, A, B}) edge12OUT≡ Xedge21OUT≡ ◇X a1OUT≡ A2OUT≡ a b1OUT≡ B2OUT≡ b otherwise ≡ False A a a B b B b b b u ⊧ a → ◇b <u,2> ⊧ A→◇b u ⊧ A2OUT→ ( edge21OUT [b1OUT] ∨ edge22OUT [b2OUT] )

30. Example • Mutual structural recursion (without accumulating parameters) can be dealt with. • For the detail of structural recursion over graphs, see [Buneman, Fernandez & Suciu 00] • fun ev( x ) = od(x) • fun ev( x ) = od(x) • fun od( x ) = ev(x) • fun od( x ) = ev(x) 1 2 a A A 1 b B 3 a A 3 4 b B B edge12OUT≡ a ∧ X edge23OUT≡ a ∧ ◇X edge13OUT≡ a ∧ ◇X edge31OUT≡ b ∧ ◇X edge34OUT≡ b ∧ X edge41OUT≡ b ∧ ◇X

31. Pre-Image Computation • invk ( False, Δ ) = False • invk ( ￢φ , Δ ) = ￢ invk ( φ, Δ ) • invk ( φ1∨φ2, Δ ) = invk ( φ1, Δ ) ∨ invk ( φ2, Δ ) • invk ( σ, Δ ) = σkOUT • invk ( ◇φ, Δ ) = ∨j∈{1..n}edgekjOUT [X / invj(φ, Δ)] • invk ( Y, Δ ) = Yk if k∈S • invk ( Y, Δ ) = μYk. invk( φ, Δ[Y→<S∪{k},φ>] ) where (S,φ) = Δ(Y) • invk ( μY.φ , Δ ) = μYk. invk( φ, Δ[Y→<{k},φ>] ) Thm: fT(G), <v,k> ⊧ φ iff G,v ⊧ invk(φ, {})

32. Example edge11OUT≡ edge12OUT≡ edge21OUT≡ edge22OUT≡ ◇X a1OUT≡ a2OUT≡ a • f(G), <v,1> ⊧ μY. (a ∧ ◇Y) • G, v ⊧ μY1. inv1( a ∧ ◇Y ) • G, v ⊧ μY1. a ∧ (◇inv1(Y)∨ ◇inv2(Y)) • G, v ⊧ μY1. a ∧ (◇Y1 ∨ ◇μY2.inv2(a∧◇Y)) • G, v ⊧ μY1. a ∧ (◇Y1 ∨ ◇μY2. a∧(◇inv1(Y)∨◇inv2(Y)) • G, v ⊧ μY1. a ∧ (◇Y1 ∨ ◇μY2. a∧(◇Y1∨◇Y2)) Open Question: can inv(μ) be shorterthan (n-1)!+1 ?

33. Some Useful Results Construction is analogous to inv(φ). Theorem:Modal-μ Definable Transduction is closed under composition. Theorem: Modal-μ Definable Transduction ⇔ MSO Definable & Bisimulation-Invariant. • It is known that Bisimulation-Invariant MSO transduction isequal to structural recursion [Colcombet & Löding04].

34. Conclusion • Modal-μ Definable Transduction • Pre-Image of a modal-μ sentence is computable • Structural recursion is expressible • (Not in the talk) • Node-erasing transformations • Edge-labeled graphs • Transformations with multiple inputs/outputs • Future Work • Implementation • Addition of Backward Modality • ( G,v ⊧◆φ iff there’s (w,v)∈E s.t. G,w ⊧ φ ) • Syntactic necessary condition for edgeOUT • More concise formula for inv(μY.φ)

35. References [Trakhtenbrot 50] Impossibility of an Algorithm for the Decision Problem for Finite Classes • Satisfiability of FO on graphs is undecidable [Meyer 74] Weak monadic second order theory of successor is not elementary-recursive • Satisfiability of MSO on finite strings is Non-Elementary [Robertson 74] Structure of Complexity in the Weak Monadic Second-Order Theories of the Natural Numbers • Satisfiability of FO[<] on finite strings is Non-Elementary [Lander 77] The Computational Complexity of Provability in Systems of Propositional Modal Logic • Satisfiability of Modal Logic on graphs is PSPACE-complete [Emereson & Jutla 88] The Complexity of Tree Automata and Logics of Programs • Satisiability of Modal-μ on graphs is EXPTIME-complete [van Benthem 86] Essays in Logical Semantics • FO ∩ Bisim = Modal [Janin & Walukiewicz 96] On the Expressive Completeness of the Propositional mu-Calculus with Respect to Monadic Second Order Logic • MSO ∩ Bisim = Modal-μ [Colcombet & Löding 04] On the Expressiveness of Deterministic Transducers over Infinite Trees • MSO-Definable Graph Transduction ∩ Bisim = Structural Recursion [Courcelle 94] Monadic Second-Order Definable Graph Transductions: A Survey • On MSO-Definable Transduction